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Stéphane Seuret

Publications and source records attributed to Stéphane Seuret.

At least 19 recordsLinked to original sources
arXiv2026

The multifractal nature of a parametrized family of von Koch functions

In a famous paper published in 1904, Helge von Koch introduced the curve that still serves nowadays as an iconic representation of fractal shapes. In fact, von Koch's main goal was the construction of a continuous but nowhere differentiable function, very similar to the snowflake, using elementary geometric procedures, and not analytical formulae. We prove that a parametrized family of functions (including and) generalizing von Koch's example enjoys a rich multifractal behavior, thus enriching the class of historical mathematical objects having surprising regularity properties. The analysis relies on the study of the orbits of an underlying dynamical system and on the introduction of self-similar measures and non-trivial iterated functions systems adapted to the problem.

math.CA↗
arXiv2026

Traces of functions in Besov spaces in Gibbs environment

This paper investigates the traces of functions belonging to the inhomogeneous Besov spaces B $ξ$ p,q , where $ξ$ is a product of capacities defined as powers of Gibbs measures. We first establish that the traces of functions in B $ξ$ p,q along affine hyperplanes belong to another inhomogeneous Besov space. Furthermore, we derive an upper bound for the singularity spectrum of the traces of all functions in B $ξ$ $\infty$,q . This bound is then refined for a prevalent set of functions in B $ξ$ $\infty$,q , for which we explicitly compute the singularity spectrum of their traces. Notably, our analysis reveals that the regularity properties of these affine traces are highly sensitive to the choice of the hyperplane along which the trace is taken.

math.FA↗
arXiv2026

A non-vanishing property for tensor products of wavelets

We prove that, given a wavelet $ψ$, it is possible to choose some multi-integers $(p_j=(p_{j,1},...,p_{j,d}))_{j \in \mathbb{Z}} \in \mathbb{Z}^d$ such that, for every $x=(x_1,...,x_d) \in \mathbb{R}^d$, for infinitely many integers $j$, the tensorized wavelet $\prod_{i=1}^d ψ(2^j x_i-p_{j,i})$ does not vanish at $x$. This non-vanishing property is essential for analyzing some generic regularity properties in certain Sobolev and Besov spaces. The proof relies on an assumption regarding the zeros of $ψ$, which we numerically verify for the first Daubechies wavelets.

math.FA↗
arXiv2025

Hitting Probabilities and the Ekstr{ö}m-Persson conjecture

We consider the Ekst\''om-Persson conjecture concerning the value of the Hausdorff dimension of random covering sets formed by balls with radii $(k^{-α})_{k=1}^\infty$ and centres chosen independently at random according to an arbitrary Borel probability measure $μ$ on $\mathbb{R}^d$. The conjecture has been solved positively in the case $\frac 1α\le \overline{\dim}_H μ$, where $\overline{\dim}_H μ$ stands for the upper Hausdorff dimension of $μ$. In this paper, we develop a new approach in order to answer the full conjecture, proving in particular that the conjectured value is only a lower bound for the dimension. Our approach opens the way to study more general limsup sets, and has consequences on the so-called hitting probability questions. For instance, we are able to determine whether and what part of a deterministic analytic set can be hit by random covering sets formed by open sets.

math.PR↗
arXiv2024

On the multivariate multifractal formalism: examples and counter-examples

In this article, we investigate the bivariate multifractal analysis of pairs of Borel probability measures. We prove that, contrarily to what happens in the univariate case, the natural extension of the Legendre spectrum does not yield an upper bound for the bivariate multifractal spectrum. For this we build a pair of measures for which the two spectra have disjoint supports. Then we study the bivariate multifractal behavior of an archetypical pair of randomly correlated measures, which give new, surprising, behaviors, enriching the narrow class of measures for which such an analysis is achieved.

math.MG↗
arXiv2023

Sparse sampling and dilation operations on a Gibbs weighted tree, and multifractal formalism

In this article, starting from a Gibbs capacity, we build a new random capacity by applying two simple operators, the first one introducing some redundancy and the second one performing a random sampling. Depending on the values of the two parameters ruling the redundancy and the sampling, the new capacity has very different multifractal behaviors. In particular, the multifractal spectrum of the capacity may contain two to four phase transitions, and the multifractal formalism may hold only on a strict subset (sometimes, reduced to a single point) of the spectrum's domain.

math.MG↗
arXiv2022

Measures, annuli and dimensions

Given a Radon probability measure $μ$ supported in $\mathbb{R}^d$, we are interested in those points $x$ around which the measure is concentrated infinitely many times on thin annuli centered at $x$. Depending on the lower and upper dimension of $μ$, the metric used in the space and the thinness of the annuli, we obtain results and examples when such points are of $μ$-measure $0$ or of $μ$-measure $1$. The measure concentration we study is related to ''bad points'' for the Poincaré recurrence theorem and to the first return times to shrinking balls under iteration generated by a weakly Markov dynamical system. The study of thin annuli and spherical averages is also important in many dimension-related problems, including Kakeya-type problems and Falconer's distance set conjecture.

math.CA↗
arXiv2022

Potential method and projection theorems for macroscopic Hausdorff dimension

The macroscopic Hausdorff dimension Dim H (E) of a set E $\subset$ R d was introduced by Barlow and Taylor to quantify a "fractal at large scales" behavior of unbounded, possibly discrete, sets E. We develop a method based on potential theory in order to estimate this dimension in R d. Then, we apply this method to obtain Marstrand-like projection theorems: given a set E $\subset$ R 2 , for almost every $θ$ $\in$ [0, 2$π$], the projection of E on the straight line passing through 0 with angle $θ$ has dimension equal to min(Dim H (E) , 1).

math.CA↗
arXiv2021

Multifractal analysis of sums of random pulses

In this paper, we determine the almost sure multifractal spectrum of a class of random functions constructed as sums of pulses with random dilations and translations. In addition, the continuity modulii of these functions is investigated.

math.CA↗
arXiv2021

Besov spaces in multifractal environment and the Frisch-Parisi conjecture

In this article, a solution to the so-called Frisch-Parisi conjecture is brought. This achievement is based on three ingredients developed in this paper. First almost-doubling fully supported Radon measures on $\R^d$ with a prescribed singularity spectrum are constructed. Second we define new \textit{heterogeneous} Besov spaces $B^{μ,p}_{q}$ and find a characterization using wavelet coefficients. Finally, we fully describe the multifractal nature of typical functions in the function spaces $B^{μ,p}_{q}$. Combining these three results, we find Baire function spaces in which typical functions have a prescribed singularity spectrum and satisfy a multifractal formalism. This yields an answer to the Frisch-Parisi conjecture.

math.FA↗
arXiv2017

Multifractal properties of typical convex functions

We study the singularity (multifractal) spectrum of continuous convex functions defined on $[0,1]^{d}$. Let $E_f({h}) $ be the set of points at which $f$ has a pointwise exponent equal to $h$. We first obtain general upper bounds for the Hausdorff dimension of these sets $E_f(h)$, for all convex functions $f$ and all $h\geq 0$. We prove that for typical/generic (in the sense of Baire) continuous convex functions $f:[0,1]^{d}\to \mathbb{R} $, one has $\dim E_f(h) =d-2+h$ for all $h\in[1,2],$ and in addition, we obtain that the set $ E_f({h} )$ is empty if $h\in (0,1)\cup (1,+\infty)$. Also, when $f$ is typical, the boundary of $[0,1]^{d}$ belongs to $E_{f}({0})$.

math.CA↗
arXiv2016

Multifractal analysis for the occupation measure of stable-like processes

In this article, we investigate the local behaviors of the occupation measure $μ$ of a class of real-valued Markov processes M, defined via a SDE. This (random) measure describes the time spent in each set A $\subset$ R by the sample paths of M. We compute the multifractal spectrum of $μ$, which turns out to be random, depending on the trajec-tory. This remarkable property is in sharp contrast with the results previously obtained on occupation measures of other processes (such as L{é}vy processes), since the multifractal spectrum is usually determinis-tic, almost surely. In addition, the shape of this multifractal spectrum is very original, reflecting the richness and variety of the local behaviors. The proof is based on new methods, which lead for instance to fine estimates on Hausdorff dimensions of certain jump configurations in Poisson point processes.

math.DS↗
arXiv2015

Random sparse sampling in a Gibbs weighted tree

Let $μ$ be the geometric realization on $[0,1]$ of a Gibbs measure on $Σ=\{0,1\}^{\mathbb{N}}$ associated with a Hölder potential. The thermodynamic and multifractal properties of $μ$ are well known to be linked via the multifractal formalism. In this article, the impact of a random sampling procedure on this structure is studied. More precisely, let $\{I_w\}_{w\in Σ^*}$ stand for the collection of dyadic subintervals of $[0,1]$ naturally indexed by the set of finite dyadic words $Σ^*$. Fix $η\in(0,1)$, and a sequence $(p_w)_{w\in Σ^*}$ of independent Bernoulli variables of parameters $2^{-|w|(1-η)}$ ($|w|$ is the length of $w$). We consider the (very sparse) remaining values $\widetildeμ=\{μ(I_w): w\in Σ^*, p_w=1\}$. We prove that when $η<1/2$, it is possible to entirely reconstruct $μ$ from the sole knowledge of $\widetildeμ$, while it is not possible when $η>1/2$, hence a first phase transition phenomenon. We show that, for all $η\in (0,1)$, it is possible to reconstruct a large part of the initial multifractal structure of $μ$, via the fine study of $\widetildeμ$. After reorganization, these coefficients give rise to a random capacity with new remarkable scaling and multifractal properties: its $L^q$-spectrum exhibits two phase transitions, and has a rich thermodynamic and geometric structure.

math-ph↗
arXiv2015

Multifractal Analysis of functions on Heisenberg and Carnot Groups

In this article, we investigate the pointwise behaviors of functions on the Heisenberg group. We find wavelet characterizations for the global and local Hölder exponents. Then we prove some a priori upper bounds for the multifractal spectrum of all functions in a given Hölder, Sobolev or Besov space. These upper bounds turn out to be optimal, since in all cases they are reached by typical functions in the corresponding functional spaces. We also explain how to adapt our proof to extend our results to Carnot groups.

math.FA↗
arXiv2014

Local $L^2$-regularity of Riemann's Fourier series

We are interested in the convergence and the local regularity of the lacunary Fourier series $F_s(x) = \sum_{n=1}^{+\infty} \frac{e^{2iπn^2 x}}{n^s}$. In the 1850's, Riemann introduced the series $F_2$ as a possible example of nowhere differentiable function, and the study of this function has drawn the interest of many mathematicians since then. We focus on the case when $1/2<s\leq 1$, and we prove that $F_s(x)$ converges when $x$ satisfies a Diophantine condition. We also study the $L^2$- local regularity of $F_s$, proving that the local $L^2$-norm of $F_s$ around a point $x$ behave differently around different $x$, according again to Diophantine conditions on $x$.

math.FA↗
arXiv2013

Quantitative recurrence properties in conformal iterated function systems

Let $Λ$ be a countable index set and $S=\{ϕ_i: i\in Λ\}$ be a conformal iterated function system on $[0,1]^d$ satisfying the open set condition. Denote by $J$ the attractor of $S$. With each sequence $(w_1,w_2,...)\in Λ^{\mathbb{N}}$ is associated a unique point $x\in [0,1]^d$. Let $J^\ast$ denote the set of points of $J$ with unique coding, and define the mapping $T:J^\ast \to J^\ast$ by $Tx= T (w_1,w_2, w_3...) = (w_2,w_3,...)$. In this paper, we consider the quantitative recurrence properties related to the dynamical system $(J^\ast, T)$. More precisely, let $f:[0,1]^d\to \mathbb{R}^+$ be a positive function and $$R(f):=\{x\in J^\ast: |T^nx-x|<e^{-S_n f(x)}, \ {\text{for infinitely many}}\ n\in \mathbb{N}\},$$ where $S_n f(x)$ is the $n$th Birkhoff sum associated with the potential $f$. In other words, $R(f)$ contains the points $x$ whose orbits return close to $x$ infinitely often, with a rate varying along time. Under some conditions, we prove that the Hausdorff dimension of $R(f)$ is given by $\inf\{s\ge 0: P(T, -s(f+\log |T'|))\le 0\}$, where $P$ is the pressure function and $T'$ is the derivative of $T$. We present some applications of the main theorem to Diophantine approximation.

math.DS↗
arXiv2013

Measures and functions with prescribed homogeneous multifractal spectrum

In this paper we construct measures supported in $[0,1]$ with prescribed multifractal spectrum. Moreover, these measures are homogeneously multifractal (HM, for short), in the sense that their restriction on any subinterval of $[0,1]$ has the same multifractal spectrum as the whole measure. The spectra $f$ that we are able to prescribe are suprema of a countable set of step functions supported by subintervals of $[0,1]$ and satisfy $f(h)\leq h$ for all $h\in [0,1]$. We also find a surprising constraint on the multifractal spectrum of a HM measure: the support of its spectrum within $[0,1]$ must be an interval. This result is a sort of Darboux theorem for multifractal spectra of measures. This result is optimal, since we construct a HM measure with spectrum supported by $[0,1] \cup {2}$. Using wavelet theory, we also build HM functions with prescribed multifractal spectrum.

math.CA↗